Distribution

logGamma(x: number): number

Γ(x) overflows a double for x greater than about 171, so the log form is used throughout — every consumer here needs ratios of gamma functions, which become differences of logs and stay in range.

Accurate to roughly 15 significant digits across the positive reals. Values of x below 0.5 are handled by the reflection formula Γ(x)Γ(1−x) = π / sin(πx).

logGamma(1)   // 0        (Γ(1) = 1)
logGamma(5)   // ≈ 3.1781 (Γ(5) = 24)
logGamma(0.5) // ≈ 0.5724 (Γ(½) = √π)

regularizedIncompleteBeta(a: number, b: number, x: number): number

This is the CDF of the Beta(a, b) distribution, and the workhorse behind the Student’s t, F, and binomial tail probabilities.

The continued fraction converges quickly for x < (a+1)/(a+b+2); outside that range the symmetry I_x(a, b) = 1 − I_{1−x}(b, a) is applied first.

regularizedIncompleteBeta(1, 1, 0.4)  // 0.4   — Beta(1,1) is uniform
regularizedIncompleteBeta(1, 2, 0.5)  // 0.75  — 1 − (1 − x)²

studentTCdf(df: number, t: number): number

Computed from the regularised incomplete beta function via P(|T| > t) = I_{df/(df+t²)}(df/2, ½).

Degrees of freedom are the slowly-varying parameter, so they come first — matching round(precision, value) and quantile(p, numbers) elsewhere in this package, and making the function partial-application friendly.

studentTCdf(1, 0)   // 0.5   — symmetric about zero for every df
studentTCdf(1, 1)   // 0.75  — df = 1 is Cauchy: ½ + atan(t)/π
studentTCdf(10, -2) // ≈ 0.0367

studentTQuantile(df: number, p: number): number

This is what turns a confidence level into a multiplier — a 95% interval on a coefficient is estimate ± studentTQuantile(df, 0.975) · standardError.

Found by bisection on {@link studentTCdf}, which is monotonic, so convergence is guaranteed. Cost is irrelevant in practice: an interval band needs one quantile for the whole curve, not one per point, because the multiplier depends only on df and the level.

studentTQuantile(10, 0.975)  // ≈ 2.228 — the classic 5% two-tailed multiplier
studentTQuantile(1, 0.75)    // 1       — df = 1 is Cauchy: tan(pi(p - 1/2))
studentTQuantile(30, 0.5)    // 0       — symmetric about zero

studentTTwoTailedP(df: number, t: number): number

This is the quantity reported alongside a regression coefficient — the probability of observing a coefficient at least this far from zero if the true coefficient were zero.

studentTTwoTailedP(10, 2.228)  // ≈ 0.05  — the classic 5% critical value
studentTTwoTailedP(10, 0)      // 1       — no evidence of any effect